In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a
polytope (for example, a
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
or a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all o ...
) or a
tiling is isotoxal () or edge-transitive if its
symmetries act
transitively on its
edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a
translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
,
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
, and/or
reflection that will move one edge to the other, while leaving the region occupied by the object unchanged.
Isotoxal polygons
An isotoxal polygon is an even-sided i.e.
equilateral polygon, but not all equilateral polygons are isotoxal. The
duals
''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers.
Track listing
:* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, P ...
of isotoxal polygons are
isogonal polygon
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of fac ...
s. Isotoxal
-gons are
centrally symmetric
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is inv ...
, so are also
zonogons.
In general, an isotoxal
-gon has
dihedral symmetry. For example, a
rhombus is an isotoxal "
×
-gon" (quadrilateral) with
symmetry. All
regular polygons (
equilateral triangle,
square, etc.) are isotoxal, having double the minimum symmetry order: a regular
-gon has
dihedral symmetry.
An isotoxal
-gon with outer internal angle
can be labeled as
The inner internal angle
may be greater or less than
degrees, making convex or concave polygons.
Star polygons can also be isotoxal, labeled as
with
and with the
greatest common divisor where
is the
turning number or
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
.
[''Tilings and Patterns'', Branko Gruenbaum, G.C. Shephard, 1987. 2.5 Tilings using star polygons, pp. 82-85.] Concave inner vertices can be defined for
If
then
is "reduced" to a compound
of
rotated copies of
Caution: The vertices of
are not always placed like those of
whereas the vertices of the regular
are placed like those of the regular
A set of "
uniform tilings
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, s ...
", actually
isogonal tilings using isotoxal polygons as less symmetric faces than regular ones, can be defined.
Isotoxal polyhedra and tilings
Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive).
Quasiregular polyhedra, like the
cuboctahedron and the
icosidodecahedron, are isogonal and isotoxal, but not isohedral. Their duals, including the
rhombic dodecahedron and the
rhombic triacontahedron, are isohedral and isotoxal, but not isogonal.
Not every
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all o ...
or 2-dimensional
tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
constructed from
regular polygons is isotoxal. For instance, the
truncated icosahedron
In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares ...
(the familiar soccerball) is not isotoxal, as it has two edge types: hexagon-hexagon and hexagon-pentagon, and it is not possible for a symmetry of the solid to move a hexagon-hexagon edge onto a hexagon-pentagon edge.
An isotoxal polyhedron has the same
dihedral angle for all edges.
The dual of a convex polyhedron is also a convex polyhedron.
The dual of a non-convex polyhedron is also a non-convex polyhedron.
(By contraposition.)
The dual of an isotoxal polyhedron is also an isotoxal polyhedron. (See the
Dual polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the oth ...
article.)
There are nine
convex isotoxal polyhedra: the five (
regular)
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s, the two (
quasiregular) common cores of dual Platonic solids, and their two duals.
There are fourteen non-convex isotoxal polyhedra: the four (regular)
Kepler–Poinsot polyhedra, the two (quasiregular) common cores of dual Kepler–Poinsot polyhedra, and their two duals, plus the three quasiregular ditrigonal (3 , ''p'' ''q'') star polyhedra, and their three duals.
There are at least five isotoxal polyhedral compounds: the five
regular polyhedral compounds; their five duals are also the five regular polyhedral compounds (or one chiral twin).
There are at least five isotoxal polygonal tilings of the Euclidean plane, and infinitely many isotoxal polygonal tilings of the hyperbolic plane, including the Wythoff constructions from the
regular hyperbolic tilings , and non-right (''p q r'') groups.
See also
*
Table of polyhedron dihedral angles
Table may refer to:
* Table (furniture), a piece of furniture with a flat surface and one or more legs
* Table (landform), a flat area of land
* Table (information), a data arrangement with rows and columns
* Table (database), how the table data ...
*
Vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of fa ...
*
Face-transitive
*
Cell-transitive
References
* Peter R. Cromwell, ''
Polyhedra'', Cambridge University Press 1997, , p. 371 Transitivity
* (6.4 Isotoxal tilings, 309-321)
*
{{polygons
Polyhedra
4-polytopes